(C) 2000 OPA (Overseas Publishers Association) N.V.
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Printed in Singapore.
J. ofInequal. & Appl., 2000, Vol. 5, pp. 191-199
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Spherical Derivative of Meromorphic
Function with Image of Finite
Spherical Area
SHINJI YAMASHITA*
Department of Mathematics, Tokyo Metropofitan University, Minami-Osawa,
Hachioji, Tokyo 192-0397, Japan
(Received 21 April 1999; Revised 18 May 1999)
Let f be a domain in the complex plane C with the Poincar6 metric P(z)ldz[ which is
Idzl/(1- Izl ) if f is the open unit disk. Suppose that the Riemann sphere CU {o} of
radius 1/2, so that it has the area 7r and let 0 < fl < 7r. Let a,(z), z E r, be the supre-
.
mum of the spherical derivative If’(z)l/(1 / If(z)l 2) of f meromorphic in f such that
the spherical area of the imagef(f) c C U {o} is not greater than Then
ZEfl.
The equality holds if and only if f is simply connected.
Keywords: Chordal distance; Spherical derivative; Spherical area; Poincar6 metric
1991 Mathematics Subject
Classification: Primary 30D32; Secondary 30F99
1. INTRODUCTION
The complex plane C R2, together with the point o at infinity, is
identified with the Riemann sphere C* of radius 1/2 touching C from
above at the origin with the aid of the stereographic projection viewed
from the north pole of C*. The sphere C* =CU {o} is the metric
subspace of the Euclidean space R 3, so that it has the distance X(z, w)
* E-mail: yamashin@comp.metro-u.ac.jp.
191
S. YAMASHITA
192
which can be expressed by
X(z, w)
.......
z
z, w
c;
C;
x(, )
0.
The spherical area of a set E C C* is given by the double integral
A(E)
Jf
dx dy
\{}
(1 + Izl2) 2’
x + iy;
z
here E is identified with its projection. For example, the spherical cap
E(a,r) {z E C*; X(z,a) < r} of center a E C* and radius, r, 0 < r < 1, has
the area 7rr 2, so that A(C)--A(E(O, 1))=Tr. Actually, A(E(a,r))=
Let A/[ (gt) be the family of all the meromorphic functions in a domain
f C C; the constant function o is regarded as a member of A/[ (gt). The
spherical derivative off A4 (f) at z 9t is defined by
f#(z)
hence
f(z)
lim
Iw-l-0
X(f(w),f(z))
Iw-zl
.
f#(z) If’(z)[/(1 + l/(z)l =) if f(z) :/: o and f#(z)=
Note that (o)#(z) 0. The set ofw C* assumed byf
if
at least once in f is denoted byf(f); hencef(f) c C* and w Ef(gt) if and
only if w =f(z) for a z 9t. For a constant/3, 0 </3 < 7r, we let .T’(gt,/3)
be the set of all f/(gt) such that A(f(f’t))</3. Note that
A(f(f’t)) <_ f#(z) dx dy; the right-hand side integral may possibly
fff
be +.
We suppose that 9t is hyperbolic, namely, C\gt contains at least two
points. Let 4 be a universal covering projection from D {w; [w[ < }
onto f, b Proj(f) in notation; b is holomorphic and b’ is zero-free. The
Poincar6 density P is then the function in ft defined by
P(z)=(l_l)l’w’2"’-"w
I)lq, "’’
z
SPHERICAL DERIVATIVE OF MEROMORPHIC FUNCTION
193
where z
z
b(w); the choice of E Proj(f) and w is immaterial as far as
b(w) is satisfied.
We begin with
For each f .T’(f, /) for hyperbolic f, the estimate holds:
THEOREM
f#(z) <_
Tr-/3 /3p(z
(1.1)
f. If the equality holds in (1.1) at a point z f, then f maps f
univalently onto a spherical cap. Conversely if f is mapped by a
meromorphic function f .(f,/) univalently onto a spherical cap, then
there exists exactly one point z fl where the equality holds in (1.1).
at each z
THEOREM 2 The family U(f, fl)for hyperbolic f is compact. Namely,
givenfn E .T(f,/),n 1,2,..., wehaveasubsequence {fnj} of {fn} andan
f .T’(f,/) such that
maxX( f:(z),. .,.
zE
f(z))
0 as
n
--+
c
(1.2)
for each compact set E (in C) comprised in f.
Set
ca,(z)
sup{f#(z);f ’(f, /3) },
z C f.
It then follows from Theorem 2 that the supremum is the maximum;
f#(z) c,(z) for anf E .T’(f,/). For this extremal function we set
g(w)
f(w) -f(z)
if f (z) # oo;
eiargf’(z)(1 + f(z)f(w))’
if f (z) c.
eiarg(1/f)’(z)f (w)
Then g C br(f,/3) with g(z)=0 and
extremal.
THEOREM 3
O<g’(z)=cn,5(z). Again, g is
If f is hyperbolic, then
<_
(1.3)
S. YAMASHITA
194
at each z E f. If the equality holds in (1.3) at apoint z f, then f is simply
connected. If f is simply connected, then the equality holds in (1.3)
everywhere in f.
We omit the detailed proof of Theorem 3 because we have only to
apply Theorem 2 to an extremal function.
2. PROOF OF THEOREM 1
LEMMA
For f .(D,/3) the inequality holds:
f#(O) 2 <
(2.1)
The equality holds in (2.1) if and only if
f(z)
=_
az+b
zD,
baz
(2.2)
where a C and b C* are constants with
Readf(z)
1/(az) in case b
in [2, Lemma I, (2)]. The
This lemma is Dufresnoy’s, the case r0
equality condition in the present "if and only if" form is obtained in the
similar manner as in [1, pp. 219-220]. Note that Dufresnoy adopted
the unit sphere of center at the origin in 1 3 as the Riemann sphere, so that
we need obvious changes.
Proof of Theorem We choose
observe thatfo q .T’(D,/3). Since
E Proj(f) with z
(fo q)#(O) =f#(z)l’(0)l =f#(z)/Pn(z),
(0) and we
SPHERICAL DERIVATIVE OF MEROMORPHIC FUNCTION
195
the inequality (1.1) is a consequence of (2.1). The equality holds in (1.1) at
z if and only if
(f o dp)(w)
aw+b
ba- w’
w E D,
where lal x/C//(Tr /). Hencefis univalent in 9t, so that is univalent
in O. The image f(f) is {(aw + b)/(1- baw); w D} which is the
image of the cap E(0, lal/v/(1 / la12)) by the rotation of C*" T(()
(( + b)/(1 b(), so thatf(ft) is a spherical cap.
Suppose that f E ’(f,/3) maps Ft univalently onto a spherical cap
E(a, X/-/Tr), a C* and set p V//3/(Tr-/3). Then
(w) f_,
(.pw
is in Proj(ft). Since
-
f#(O(w))
p(1 -[wl 2)
Pa((w))
+ p2[w12
it follows that f#(O(w))= pPa((w)) if and only if w =0. Hence the
equality holds in (1.1) at exactly one point (0), the inverse of a byf.
If z
w, then
X(z, w) <_ arctan
+w
fv
dX(),
where dX(()= [d(l/(1 + I12) and F is the projection of the shorter of
we
the great circle passing through, and bisected by z, w; in case z#
have many I’. Here 0 < arctan p < 7r/2 for 0 < p <
Suppose that/3 A(f) < 7r for a domain ft c C. We then have
dX(z) _<
/.Tr -/3/3pa(z)ldzl,
zErO.
More precisely,
-t-Izl 2
<i.TrA((f,t).pa(z)
z
e
a.
(2.3)
S. YAMASHITA
196
The equality holds in (2.3) at a point z E 9t if and only iff is a spherical cap
-a(’
( D
where a E C and ]al v/A(f)/(r A(f)). For the proof we have only
to follow that of Theorem withf(O (,f E .’(f,/3).
3. PROOF OF THEOREM 2
We begin with the case f D. Sincef # off E .T’(D,/3) is bounded by
7r--/3 1--r2
{[z[ < r}, 0 < r < 1, it follows that ’(D,/3) is equicontinuous in X on
{]z[ < r}. Hence Or(D,/3) is normal in that sense that given {f} c
-(D,/3), we have a subsequence {fnj} C {fn} andfE AA(D) such that
on
(1.2) holds for each compact set E comprised in D. To prove that
f E U(D,/3) we may suppose thatfis nonconstant.
For simplicity we suppose that
forfn E (D,/3), n 1,2,... We shall then prove that for each b E D we
have r r(b) > 0 and a natural number N N(b) such thatf(A) cfn(ft)
for all n > N, where
/x
{z;
Iz- bl < r} c D.
(3.1)
Then for each compact set K C D we have n such thatf(K)cfn(ft), so
that A(f(K)) < 3, whencef
We first suppose thatf(b)- o. Then we have rl > 0 and a constant
M1 >0 such that f is pole-free and bounded, Ifl < M, on /X
{z;Iz-b[ _< r} c D. We then find a constant ME > M1 and a natural
number N1 such that f is pole-free and bounded, ILl< M=, on/x for
SPHERICAL DERIVATIVE OF MEROMORPHIC FUNCTION
n > N1. Hence
max If(z)
zA
-f(z)[
0 as n
.
197
(3.2)
Consequently, for A of (3.1) with 0 < r < rl, we have
sup I(z)-f’(z)l
0 as n
o.
(3.3)
zEA
To prove that the present A is the requested we suppose that there
exists q E Asuch thatp =f(q) fnj(f) forinfinitelymany nj, nl < n2 <...
Choose R > 0 such that
{z; Iz-ql
<- R} C A
and pf({z; 0 <
Iz- ql < R}),
and set
{z;
c
Iz- q[- R}.
The argument principle then shows that
[
f’(()
dr.
The right-hand side integral is, with the aid of (3.2) and (3.3), the limit
if
lim
nj-ofn(f)-fn(q) d=O.
This is a contradiction.
In the casef(b) c we consider {1/fn} C ’(f,/3) with 1/fE
and arrive at a contradiction again.
For general we fix b Proj(). Then, for each compact set Ein we
may find a compact set E1 C D such that b(E1)= E. Furthermore, q is
automorphic with respect to the universal covering transformation
Since fn o q (D, fl), we have a subseq5 b o T, T
group
quence {fn } of {f} and g ’(D,/3) such thatf, o b converges to g on
each compact set in D. Since g is then automorphic with respect to we
havef E A//(f) such thatfo 4=g and thisfis the requested.
.
,
S. YAMASHITA
198
4. CONFORMAL INVARIANT
C,,()
Let E be another domain in C and let f be holomorphic in f with
f(f) c E. Then
c,(f(z))lf’(z)l < cn,(z),
z
t.
In particular, iffis univalent andf(f)= E, then
c,(f(z))lf’(z)l- ca,(z),
z
t,
so that c,(z)ldz[ is conformally invariant.
Let Bz(f) be the family of allfholomorphic, bounded, If[ < 1, in f,
and further, f (z) O, z E f. Then
sup{lf’(z)l; fE/3z(f)}
"7(z)
is called the analytic capacity of f at z. Then 3’(z) is the maximum
7,(z) [f’(z)l =f’(z) for a uniquef E Bz(f) called the Ahlfors function
of f at z. See [3, p. 110]. Since Bz(f) c )r(f, 7r/2), it follows that
"[f’t(Z)
Cf’t,Tr/2(Z),
Z
".
On the other hand, it follows from (1.3) that
n(z) <_ e.(z),
z
Iff E )r(f, fl), then C*\f(F) is of positive spherical area, so that this
set is of positive capacity. Hence f is of uniformly bounded characteristic in f; see [5, Theorem 1] and also [4,6]. Suppose that each function
of uniformly bounded characteristic in f is constant. Thenf ’(f,/3)
is a constant.
References
[1] J. Dufresnoy, Sur les domaines couverts par les valeurs d’une fonction m6romorphe ou
alg6bro’ide. Ann. Sci. Ecole Norm. Sup. 58 (1941), 179-259.
[2] J. Dufresnoy, Sur l’aire sph6rique d6crite par les valeurs d’une fonction m6romorphe.
Bull. Sci. Math. 65 1941), 214- 219.
[3] S.D. Fisher, Function Theory on Planar Domains. John Wiley, New York, 1983.
SPHERICAL DERIVATIVE OF MEROMORPHIC FUNCTION
199
[4] S. Yamashita, Functions of uniformly bounded characteristic. Ann. Acad. Sci. Fenn.,
Ser. A I Math. 7 (1982), 349-367.
[5] S. Yamashita, Exceptional sets, subordination, and functions of uniformly bounded
characteristic. Math. Scand. 54 (1984), 242-252.
[6] S. Yamashita, Functions of uniformly bounded characteristic on Riemann surfaces.
Trans. Amer. Math. Soc. 288 f1985, 395-412.